marybeth,
Feb-Mar is a great time to visit Egypt. We took a 12 day tour there (10 full days on the ground) in February 2019 with Trafalgar Tours and loved it. Two full days in Cairo, a seven day Nile cruise, day trip (by plane) from Aswan to Abu Simble, followed by another full day in Cairo before returning home. We visited all the places I hoped for. The tour had 38 people and a superb guide. We did not feel like part of a mob, as the tour members were a fun and congenial lot. Accommodations, our boat, food, activities and tours were all great and it was one of the best times I have had traveling. (The fact that Egypt was my number one bucket list item didn't hurt.) And the price was in the moderate range...worth every penny! Your 3 weeks would give you time for Egypt AND a trip to Jordan, which everyone who has been there raves about. Or perhaps a stopover in Istanbul? It is worth consideration. Should you do go there, don't fly Lufthansa. It has gone downhill re service and comfort, IMO. I used to choose Lufthansa a lot. Turkish Air has been a good choice for us more recently, and a stopover in Istanbul could be an opportunity to see some of a fascinating country!
Just a couple of suggestions for you. Whatever you decide, have a great trip!